Observation

In the previous two examples, the functions $F(x)$ and $G(x)$ were both shown to be antiderivatives of $f(x)=15x^4+12x^2+7$. This implies that antiderivatives are not unique. Also notice that $G(x)=F(x)+71$ and adding a constant to a function doesn’t change its derivative since the derivative of a constant is zero. So once we knew that $F(x)$ was an antiderivative of $f(x)$, we didn’t need to verify that $G(x)$ was also an antiderivative of $f(x)$ since it only differs from $F(x)$ by a constant.

In general, if $F$ is an antiderivative of $f$, then $F(x)+C$, where $C$ is any constant, is also an antiderivative of $f$. Furthermore, every antiderivative of $f$ must be of the form $F(x)+C$ for some constant $C$.